To find the half-life of the radioactive substance, we use the equation:

$y = y_0 e^{-0.092t}$

Step 1: Define the half-life

The half-life ($t_{1/2}$) is the time it takes for the substance to decay to half its initial amount. Mathematically, when $y = \frac{y_0}{2}$, we solve for $t$.

$\frac{y_0}{2} = y_0 e^{-0.092t}$

Step 2: Simplify the equation

Divide both sides by $y_0$:

$\frac{1}{2} = e^{-0.092t}$

Step 3: Take the natural logarithm of both sides

$\ln\left(\frac{1}{2}\right) = -0.092t$

We know that $\ln\left(\frac{1}{2}\right) = \ln(1) - \ln(2) = -\ln(2)$. So:

$-\ln(2) = -0.092t$

$t = \frac{\ln(2)}{0.092}$

Step 4: Calculate the half-life

The natural logarithm of 2 is approximately 0.693. Substituting:

$t = \frac{0.693}{0.092} \approx 7.5 \, \text{days}$

Final Answer:

The half-life of the substance is 7.5 days.

Would you like more explanation or examples? Here are some related questions:

1. How does changing the decay constant affect the half-life?
2. What is the half-life if the decay constant is doubled?
3. Can you derive the formula for half-life in terms of the decay constant?
4. What does the value of $e$ represent in this context?
5. How would the half-life change if the decay was measured in hours instead of days?

Tip: Remember that the decay constant ($k$) is inversely proportional to the half-life.